David Cameron and Ed Miliband engaged in a war of words at prime minister's questions in a last-ditch attempt to woo votes before the local and devolved elections on Thursday.

The Labour leader, who has urged voters to send a message to the government by giving his party a "strong voice" at the polls, focused his attack on the string of "broken promises" made over the past year, telling the prime minister: "How can the public believe anything [the government is] saying at these elections tomorrow?"

Cameron fired back that people should remember the "mess" left by the Labour government and urged voters: "Don't let Labour do to your council what they did to your country."

Voters will go to the polls for elections to the devolved institutions in Scotland, Wales and Northern Ireland, councils in England and the referendum on changing Westminster's voting system.

The two leaders locked horns as Miliband focused his attack on national policies such as the forced retirement of veteran police officers and the trebling of higher education tuition fees to £9,000 a year.

He accused Cameron of being "hazy on the facts and unable to give a straight answer to a straight question" as he pressed him on police officers being "forcibly retired", citing the case of former beat officer Martin Heard, of West Midlands police, who is being asked to come back to the force as a volunteer special constable, unpaid, to fill gaps left by the cuts.

Cameron said the home secretary, Theresa May, was "trying to get more police on the beat" and accused Labour of "complete and utter hypocrisy", saying the party would not have protected police numbers.

He accused Labour-run local authorities such as Manchester city council of slashing services by 25%.

"Isn't it the case that Labour local authorities are playing politics with people's jobs?" Cameron asked.

Miliband turned his guns on tuition fees, claiming Cameron had broken a promise by previously vowing that £9,000 fees would be an exception, and also by claiming that excessive fees would be capped.

He said none of the universities planning to charge £9,000 would have their fees capped by the Office for Fair Access (Offa).

The prime minister insisted the decision "will depend on the Office for Fair Access", which would not make its decisions until July, before reminding him that tuition fees were first introduced by a Labour government.

But he added that degrees had always cost up to £9,000 and said: "The question is, who's going to pay for them?"

Miliband taunted Cameron over coalition tensions that surfaced in the cabinet on Tuesday when the Lib Dem energy secretary, Chris Huhne, confronted Cameron and George Osborne over AV campaign leaflets that he believed smeared Nick Clegg.

"I know how the energy secretary must have felt at cabinet yesterday," Miliband said.

"Remember what they said a year ago – two parties working together in the national interest. Now what do we have? Two parties threatening to sue each other in their own interest. That's what's changed in the last year. What the public are saying is that on police cuts, on tuition fees, on the NHS, this is not what we voted for.

"Having broken so many of their promises a year ago, how can the public believe anything they are saying at these elections tomorrow?"

Cameron told him: "What this coalition has done over the last year is frozen council tax, capped immigration, lifted a million people out of income tax, introduced a pupil premium, linked the pension back to earnings, cut corporation tax and set up more academies in 10 months than the last government set up in 10 years.

"And with council elections tomorrow, people should remember the mess Labour left us in and resolve to themselves: don't let Labour do to your council what they did to your country."

With hundreds of seats expected to change hands when voters choose councillors in 279 authorities in every region of England except London, Cameron fielded questions from a string of Conservatives praising their local Tory-run councils while decrying decisions taken by neighbouring Labour-run local authorities.

"And when you look at the figures, it's not just that those Conservative councils are costing less [in council tax] for a band D property, they're also doing better on measures like recycling and other service delivery.

"So it's simply not true to say that, by cutting costs, you harm services. You've got to be effectively keeping your costs down in order to provide good services."

Tensions between the Lib Dems and the Conservatives over the referendum on the voting system surfaced in the chamber after the Lib Dems' Bob Russell told Cameron that the first past the post system produced results "that would embarrass Robert Mugabe".

Russell, the MP for Colchester, said that, at last year's general election, Tories polled 49% of the votes in Essex but won 95% of the seats. "Do you think that result was fair?" he asked.

Cameron told him: "In Colchester, everyone had one vote, it was counted once, and you won. I congratulate you. In other parts of Essex, everyone had one vote, they were all counted once and many of my honourable friends won."